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June  2017, 6(2): 155-175. doi: 10.3934/eect.2017009

Stability of ground states for logarithmic Schrödinger equation with a $δ^{\prime}$-interaction

Alex H. Ardila

Department of Mathematics, IME-USP, Cidade Universitária, CEP 05508-090, São Paulo, SP, Brazil

Received  July 2016 Revised  February 2017 Published  April 2017

In this paper we study the one-dimensional logarithmic Schrödin-\break ger equation perturbed by an attractive
$δ^{\prime}$
-interaction
$i{\partial _t}u + \partial _x^2u + {\rm{ }}{\gamma ^\prime }(x)u + u{\mkern 1mu} {\rm{Log|}}u|2 = 0,(x,t) \in \mathbb{R} \times \mathbb{R} ,$
where $γ>0$. We establish the existence and uniqueness of the solutions of the associated Cauchy problem in a suitable functional framework. In the attractive $δ^{\prime}$-interaction case, the set of the ground state is completely determined. More precisely: if $0 < γ≤ 2$, then there is a single ground state and it is an odd function; if $γ>2$, then there exist two non-symmetric ground states. Finally, we show that the ground states are orbitally stable via a variational approach.
Keywords: Logarithmic Schrödinger equation, δ'-interaction, bifurcation, stability, ground states.
Mathematics Subject Classification: 35Q51, 35Q55, 37K40, 34B37.
Citation: Alex H. Ardila. Stability of ground states for logarithmic Schrödinger equation with a $δ^{\prime}$-interaction. Evolution Equations & Control Theory, 2017, 6 (2) : 155-175. doi: 10.3934/eect.2017009
References:
[1]

R. Adami and D. Noja, Existence of dynamics for a 1-d NLS equation perturbed with a generalized point defect J. Phys. A Math. Theor. 42 (2009), 495302, 19pp. doi: 10.1088/1751-8113/42/49/495302.  Google Scholar

[2]

R. Adami and D. Noja, Nonlinearity-defect interaction: Symmetry breaking bifurcation in a NLS with δ' impurity, Nanosystems, 2 (2011), 5-19.   Google Scholar

[3]

R. Adami and D. Noja, Stability and symmetry-breaking bifurcation for the ground states of a NLS with a δ' interaction, Comm. Math. Phys., 318 (2013), 247-289.  doi: 10.1007/s00220-012-1597-6.  Google Scholar

[4]

R. Adami and D. Noja, Exactly solvable models and bifurcations: The case of the cubic NLS with a δ or a δ' interaction in dimension one, Math. Model. Nat. Phenom., 9 (2014), 1-16.  doi: 10.1051/mmnp/20149501.  Google Scholar

[5]

R. Adami, D. Noja and N. Visciglia, Constrained energy minimization and ground states for NLS with point defects, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1155-1188.  doi: 10.3934/dcdsb.2013.18.1155.  Google Scholar

[6]

S. Albeverio, F. Gesztesy, R. H∅egh-Krohn and H. Holden, Solvable Models in Quantum Mechanics Springer-Verlag, New York, 1988. doi: 10.1007/978-3-642-88201-2.  Google Scholar

[7]

J. Angulo and A. H. Ardila, Stability of standing waves for logarithmic Schrödinger equation with attractive delta potential, Indiana Univ. Math. J., to appear. Google Scholar

[8]

A.H. Ardila, Orbital stability of gausson solutions to logarithmic Schrödinger equations, Electron. J. Differential Equations, 335 (2016), 1-9.   Google Scholar

[9]

I. Bialynicki-Birula and J. Mycielski, Nonlinear wave mechanics, Ann. Phys, 100 (1976), 62-93.  doi: 10.1016/0003-4916(76)90057-9.  Google Scholar

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H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.2307/2044999.  Google Scholar

[11]

T. Cazenave, Stable solutions of the logarithmic Schrödinger equation, Nonlinear. Anal., T.M.A., 7 (1983), 1127-1140.  doi: 10.1016/0362-546X(83)90022-6.  Google Scholar

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T. Cazenave, Semilinear Schrödinger Equations Courant Lecture Notes in Mathematics, 10, American Mathematical Society, Courant Institute of Mathematical Sciences, 2003. doi: 10.1090/cln/010.  Google Scholar

[13]

T. Cazenave and P. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561.  doi: 10.1007/BF01403504.  Google Scholar

[14]

R. Fukuizumi and L. Jeanjean, Stability of standing waves for a nonlinear Schrödinger equation with a repulsive {D}irac delta potential, Discrete Contin. Dyn. Syst., 21 (2008), 121-136.  doi: 10.3934/dcds.2008.21.121.  Google Scholar

[15]

R. Fukuizumi, M. Ohta and T. Ozawa, Nonlinear Schrödinger equation with a point defect, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 837-845.  doi: 10.1016/j.anihpc.2007.03.004.  Google Scholar

[16]

R. Fukuizumi and A. Sacchetti, Bifurcation and stability for nonlinear Schrödinger equations with double well potential in the semiclassical limit, J. Stat. Phys., 145 (2011), 1546-1594.  doi: 10.1007/s10955-011-0356-y.  Google Scholar

[17]

A. Haraux, Nonlinear Evolution Equations: Global Behavior of Solutions vol. 841 of Lecture Notes in Math., Springer-Verlag, Heidelberg, 1981.  Google Scholar

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E. Hefter, Application of the nonlinear Schrödinger equation with a logarithmic inhomogeneous term to nuclear physics, Phys. Rev, 32 (1985), 1201-1204.  doi: 10.1103/PhysRevA.32.1201.  Google Scholar

[19]

R.K. Jackson and M. Weinstein, Geometric analysis of bifurcation and symmetry breaking in a {G}ross-{P}itaevskii equation, J. Stat. Phys., 116 (2004), 881-905.  doi: 10.1023/B:JOSS.0000037238.94034.75.  Google Scholar

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M. Kaminaga and M. Ohta, Stability of standing waves for nonlinear {S}chrödinger equation with attractive delta potential and repulsive nonlinearity, Saitama Math. J., 26 (2009), 39-48.   Google Scholar

[21]

C.M. Khalique and A. Biswas, Gaussian soliton solution to nonlinear Schrödinger's equation with log law nonlinearity, International Journal of Physical Sciences, 5 (2010), 280-282.   Google Scholar

[22]

E.W. Kirr, P. Kevrekidis and D. Pelinovsky, Symmetry-breaking bifurcation in the nonlinear Schrödinger equation with symmetric potentials, Comm. Math. Phys., 308 (2011), 795-844.  doi: 10.1007/s00220-011-1361-3.  Google Scholar

[23]

A. Kostenko and M. Malamud, Spectral theory of semibounded Schrödinger operators with $δ^{\prime}$-interactions, Ann. Henri Poincaré, 15 (2014), 501-541.  doi: 10.1007/s00023-013-0245-9.  Google Scholar

[24]

S. Le Coz, R. Fukuizumi, G. Fibich, B. Ksherim and Y. Sivan, Instability of bound states of a nonlinear Schrödinger equation with a dirac potential, Phys. D, 237 (2008), 1103-1128.  doi: 10.1016/j.physd.2007.12.004.  Google Scholar

[25]

E. Lieb and M. Loss, Analysis 2nd edition, vol. ~14 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.  Google Scholar

[26]

A. ~Sacchetti, Universal critical power for nonlinear Schrödinger equations with symmetric double well potential Phys. Rev. Lett. 103 (2009), 194101. doi: 10.1103/PhysRevLett.103.194101.  Google Scholar

[27]

K. Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space vol. 265 of Graduate Texts in Mathematics, Springer, Dordrecht, 2012. doi: 10.1007/978-94-007-4753-1.  Google Scholar

[28]

J. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim, 12 (1984), 191-202.  doi: 10.1007/BF01449041.  Google Scholar

[29]

K. Zloshchastiev, Logarithmic nonlinearity in theories of quantum gravity: {O}rigin of time and observational consequences, Grav. Cosmol., 16 (2010), 288-297.  doi: 10.1134/S0202289310040067.  Google Scholar

show all references

References:
[1]

R. Adami and D. Noja, Existence of dynamics for a 1-d NLS equation perturbed with a generalized point defect J. Phys. A Math. Theor. 42 (2009), 495302, 19pp. doi: 10.1088/1751-8113/42/49/495302.  Google Scholar

[2]

R. Adami and D. Noja, Nonlinearity-defect interaction: Symmetry breaking bifurcation in a NLS with δ' impurity, Nanosystems, 2 (2011), 5-19.   Google Scholar

[3]

R. Adami and D. Noja, Stability and symmetry-breaking bifurcation for the ground states of a NLS with a δ' interaction, Comm. Math. Phys., 318 (2013), 247-289.  doi: 10.1007/s00220-012-1597-6.  Google Scholar

[4]

R. Adami and D. Noja, Exactly solvable models and bifurcations: The case of the cubic NLS with a δ or a δ' interaction in dimension one, Math. Model. Nat. Phenom., 9 (2014), 1-16.  doi: 10.1051/mmnp/20149501.  Google Scholar

[5]

R. Adami, D. Noja and N. Visciglia, Constrained energy minimization and ground states for NLS with point defects, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1155-1188.  doi: 10.3934/dcdsb.2013.18.1155.  Google Scholar

[6]

S. Albeverio, F. Gesztesy, R. H∅egh-Krohn and H. Holden, Solvable Models in Quantum Mechanics Springer-Verlag, New York, 1988. doi: 10.1007/978-3-642-88201-2.  Google Scholar

[7]

J. Angulo and A. H. Ardila, Stability of standing waves for logarithmic Schrödinger equation with attractive delta potential, Indiana Univ. Math. J., to appear. Google Scholar

[8]

A.H. Ardila, Orbital stability of gausson solutions to logarithmic Schrödinger equations, Electron. J. Differential Equations, 335 (2016), 1-9.   Google Scholar

[9]

I. Bialynicki-Birula and J. Mycielski, Nonlinear wave mechanics, Ann. Phys, 100 (1976), 62-93.  doi: 10.1016/0003-4916(76)90057-9.  Google Scholar

[10]

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.2307/2044999.  Google Scholar

[11]

T. Cazenave, Stable solutions of the logarithmic Schrödinger equation, Nonlinear. Anal., T.M.A., 7 (1983), 1127-1140.  doi: 10.1016/0362-546X(83)90022-6.  Google Scholar

[12]

T. Cazenave, Semilinear Schrödinger Equations Courant Lecture Notes in Mathematics, 10, American Mathematical Society, Courant Institute of Mathematical Sciences, 2003. doi: 10.1090/cln/010.  Google Scholar

[13]

T. Cazenave and P. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561.  doi: 10.1007/BF01403504.  Google Scholar

[14]

R. Fukuizumi and L. Jeanjean, Stability of standing waves for a nonlinear Schrödinger equation with a repulsive {D}irac delta potential, Discrete Contin. Dyn. Syst., 21 (2008), 121-136.  doi: 10.3934/dcds.2008.21.121.  Google Scholar

[15]

R. Fukuizumi, M. Ohta and T. Ozawa, Nonlinear Schrödinger equation with a point defect, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 837-845.  doi: 10.1016/j.anihpc.2007.03.004.  Google Scholar

[16]

R. Fukuizumi and A. Sacchetti, Bifurcation and stability for nonlinear Schrödinger equations with double well potential in the semiclassical limit, J. Stat. Phys., 145 (2011), 1546-1594.  doi: 10.1007/s10955-011-0356-y.  Google Scholar

[17]

A. Haraux, Nonlinear Evolution Equations: Global Behavior of Solutions vol. 841 of Lecture Notes in Math., Springer-Verlag, Heidelberg, 1981.  Google Scholar

[18]

E. Hefter, Application of the nonlinear Schrödinger equation with a logarithmic inhomogeneous term to nuclear physics, Phys. Rev, 32 (1985), 1201-1204.  doi: 10.1103/PhysRevA.32.1201.  Google Scholar

[19]

R.K. Jackson and M. Weinstein, Geometric analysis of bifurcation and symmetry breaking in a {G}ross-{P}itaevskii equation, J. Stat. Phys., 116 (2004), 881-905.  doi: 10.1023/B:JOSS.0000037238.94034.75.  Google Scholar

[20]

M. Kaminaga and M. Ohta, Stability of standing waves for nonlinear {S}chrödinger equation with attractive delta potential and repulsive nonlinearity, Saitama Math. J., 26 (2009), 39-48.   Google Scholar

[21]

C.M. Khalique and A. Biswas, Gaussian soliton solution to nonlinear Schrödinger's equation with log law nonlinearity, International Journal of Physical Sciences, 5 (2010), 280-282.   Google Scholar

[22]

E.W. Kirr, P. Kevrekidis and D. Pelinovsky, Symmetry-breaking bifurcation in the nonlinear Schrödinger equation with symmetric potentials, Comm. Math. Phys., 308 (2011), 795-844.  doi: 10.1007/s00220-011-1361-3.  Google Scholar

[23]

A. Kostenko and M. Malamud, Spectral theory of semibounded Schrödinger operators with $δ^{\prime}$-interactions, Ann. Henri Poincaré, 15 (2014), 501-541.  doi: 10.1007/s00023-013-0245-9.  Google Scholar

[24]

S. Le Coz, R. Fukuizumi, G. Fibich, B. Ksherim and Y. Sivan, Instability of bound states of a nonlinear Schrödinger equation with a dirac potential, Phys. D, 237 (2008), 1103-1128.  doi: 10.1016/j.physd.2007.12.004.  Google Scholar

[25]

E. Lieb and M. Loss, Analysis 2nd edition, vol. ~14 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.  Google Scholar

[26]

A. ~Sacchetti, Universal critical power for nonlinear Schrödinger equations with symmetric double well potential Phys. Rev. Lett. 103 (2009), 194101. doi: 10.1103/PhysRevLett.103.194101.  Google Scholar

[27]

K. Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space vol. 265 of Graduate Texts in Mathematics, Springer, Dordrecht, 2012. doi: 10.1007/978-94-007-4753-1.  Google Scholar

[28]

J. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim, 12 (1984), 191-202.  doi: 10.1007/BF01449041.  Google Scholar

[29]

K. Zloshchastiev, Logarithmic nonlinearity in theories of quantum gravity: {O}rigin of time and observational consequences, Grav. Cosmol., 16 (2010), 288-297.  doi: 10.1134/S0202289310040067.  Google Scholar

Figure 1.  The graph of the curve $\mathcal{I}_{1}\cup \mathcal{I}_{2}$.
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